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A systematic framework for formulating convex failure

envelopes in multiple loading dimensions

STEPHEN K. SURYASENTANA,HARVEYJ.BURD†, BYRON W. BYRNE†and AVI SHONBERG‡

The failure envelope approach is widely used to assess the ultimate capacity of shallow foundations for

combined loading, and to develop foundation macro-element models. Failure envelopes are typically

determined by fitting appropriate functions to a set of discrete failure load data, determined either

experimentallyor numerically. However, current procedures to formulate failure envelopes tend to be ad

hoc, and the resulting failure envelopes may not have the desirable features of being convex and well-

behaved for the entire domain of interest. This paper describes a new systematic framework to

determine failure envelopes –based on the use of sum of squares convex polynomials –that are

guaranteed to be convex and well-behaved. The framework is demonstrated by applying it to three

data sets for failure load combinations (vertical load, horizontal load and moment) for shallow

foundations on clay. An example foundation macro-element model based on the proposed framework

is also described.

KEYWORDS: bearing capacity; failure; footings/foundations; numerical modelling; offshore engineering;

soil/structure interaction

INTRODUCTION

Existing failure envelope formulations

The failure envelope approach is widely used to assess the

ultimate capacity of shallow foundations for combined load-

ing. There are numerous advantages of this approach com-

pared to classical bearing capacity methods (Terzaghi, 1943;

Meyerhof, 1951; Vesic

´, 1973), as has been discussed exten-

sively in previous work (e.g. Tan, 1990; Gottardi & Butterfield,

1993; Bransby & Randolph, 1998; Martin & Houlsby, 2000;

Houlsby & Byrne, 2001). Failure envelopes can also be used to

develop plasticity-based macro-element models for shallow

foundations (e.g. Nova & Montrasio, 1991; Cassidy, 1999;

Martin & Houlsby, 2001; Bienen et al., 2006).

A failure envelope for a specific shallow foundation

concept can be determined by measuring, or computing,

a set of discrete failure load combinations for different

loading conditions (e.g. the experimental data reported in

Martin & Houlsby (2000) and the numerical data in Taiebat

& Carter (2000)). These failure envelope data are then

approximated using an appropriate mathematical formu-

lation (referred to in this paper as a ‘failure envelope

formulation’) together with an optimisation process to

achieve a best fit.

In the current paper, V,Hand Mrefer to the applied

vertical and horizontal loads, and moment, respectively.

Furthermore,

˜

H¼H=H0,

˜

M¼M=M0and

˜

V¼V=V0refer

to the normalised loads, where H

0

,M

0

,V

0

are the respective

uniaxial capacities. (Uniaxial capacity refers to the failure

load for one load component when all other load

components are maintained at zero throughout the

loading.) In the notation employed in this paper, vectors

are denoted in bold face, italic, lower case (e.g. x) and

matrices are denoted in bold face, upper case, upright font

(e.g. A).

Current failure envelope formulations can generally be

categorised into one of two groups (although the wide-

ranging literature in this area also includes a number of

formulations that do not fit into either group, e.g. Taiebat &

Carter (2000)). One group represents the failure envelope

using a sum of power functions of the applied loads. This

approach has been used to represent the ‘cigar-shaped’failure

envelopes determined from experimental data for shallow

foundations on sand (Nova & Montrasio, 1991; Gottardi &

Butterfield, 1993; Cassidy, 1999; Gottardi et al., 1999; Byrne

& Houlsby, 2001; Bienen et al., 2006) and clay (Martin &

Houlsby, 2000). Published formulations of this type are

mainly defined for planar VHM loading, although exten-

sions to six degrees-of-freedom loading have been described

by Martin (1994), Byrne & Houlsby (2005) and Bienen

et al. (2006). An example of a formulation in this group is

‘Model A’(Martin, 1994), which was developed to represent

the failure envelope for a spudcan foundation in clay. Model

A is defined as

fð

˜

H;

˜

M;

˜

VÞ¼

˜

H2þ

˜

M216

˜

V2ð1

˜

VÞ2¼0ð1Þ

although for numerical implementation, Martin & Houlsby

(2001) recommend the alternative form

fð

˜

H;

˜

M;

˜

VÞ¼ð

˜

H2þ

˜

M2Þ1=24

˜

Vð1

˜

VÞ¼0ð2Þ

The second group of failure envelope formulations, termed

‘HM-based’, has primarily been used in connection with

numerically determined failure envelope data (Feng et al.,

2014; Vulpe et al., 2014; Vulpe, 2015; Shen et al., 2016,

2017). Formulations in this group are defined in terms

of composite horizontal and moment loads, where the

composition provides a means of incorporating the influence

of additional load components within a basic HM frame-

work. An example of a formulation in this group (from

Department of Engineering Science, University of Oxford,

Oxford, UK (Orcid:0000-0001-5460-5089).

†Department of Engineering Science, University of Oxford,

Oxford, UK.

‡Ørsted Wind Power, London, UK.

Manuscript received 30 October 2018; revised manuscript accepted

28 March 2019.

Discussion on this paper is welcomed by the editor.

Published with permission by the ICE under the CC-BY 4.0 license.

(http://creativecommons.org/licenses/by/4.0/)

Suryasentana, S. K. et al.Géotechnique [https://doi.org/10.1680/jgeot.18.P.251]

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Vulpe et al. (2014)) is

fð

˜

H;

˜

MÞ¼

˜

H

aþ

˜

Mαþ2b

˜

H˜

M1¼0ð3Þ

where

˜

Hand

˜

Mare composite horizontal and moment loads

defined by

˜

H¼

˜

H=ð1

˜

V469Þand

˜

M¼

˜

M=ð1

˜

V212Þ;

and aand bare parameters (a=2·13 and b=0·09 for

˜

V

05, or a=1·63 and b=0·05 for

˜

V.05).

Formulation of foundation macro-element models

Failure envelope formulations are a convenient basis for

the development of plasticity-based macro-element models

for foundations. For example, based on experimental data,

Martin & Houlsby (2000) develop a modified version of

Model A (termed ‘Model B’, described further in the later

subsection within ‘Example applications’entitled ‘Model B’)

as the yield function for a macro-element model of a spudcan

foundation in clay.

A macro-element model typically incorporates a yield

function and a plastic potential function; these functions

govern the admissible load states and the plastic displace-

ments, respectively. The yield function may also contain

a hardening parameter that governs the evolution of the

yield surface (see Appendix for discussion on the distinction

between a yield surface and a yield function). However, the

current paper is concerned mainly with developing failure

envelope formulations to provide an approximate fit to

discrete failure load data. For simplicity, for the macro-

element formulation described later in this paper, associated

flow (i.e. where the yield function is also the plastic potential

function) and perfect plasticity (i.e. no hardening) are

adopted.

The yield function adopted in macro-element models

should ideally be convex throughout the domain and it is

insufficient to have convexity at just the admissible loads

boundary (i.e. a convex yield surface). Franchi et al. (1990)

has discussed the key differences between the yield function

and the yield surface and the theoretical relevance of the

convexity of the yield function. A summary of the charac-

teristics of convex functions and the difference between a

yield surface and a yield function is given in the Appendix.

The importance of convexity of the yield function has been

largely overlooked, especially in geotechnics, over the past

few decades. This particular issue was raised by Panteghini

& Lagioia (2014, 2018a, 2018b), who reported numerical

problems (e.g. lack of convergence in boundary value

problems) when using implicit integration algorithms with

non-convex yield functions. To avoid such problems, they

propose an approach for providing convexity to the yield

functions. For rate-independent, non-hardening/softening

materials, a convex yield function has the added benefit of

thermodynamic consistency, provided that an associated

flow rule is adopted and the yield surface contains the

origin of the load space (Ottosen & Ristinmaa, 2005).

Yield (and plastic potential) functions employed in macro-

element models must be continuous and real-valued. Ideally,

they should also be differentiable with a continuous gradient

and Hessian throughout the load domain. Furthermore,

they should not have a restricted domain, as it is not known

a priori the load states at which implicit integration algo-

rithms will require the evaluation of the yield function.

Also, functions should be avoided that have singularities

(either in the function itself or its derivatives) at certain load

states. For the purposes of the current paper, a function

with these ideal attributes is referred to as ‘well-behaved’.

It is noted that successful plasticity models can be

developed on the basis of yield functions that are not

well-behaved; the conventional Mohr–Coulomb yield func-

tion, for example, provides a common basis for soil con-

stitutive models despite not being differentiable at the ‘edges’

of the yield surface (although rounding approximations

(e.g. Sloan & Booker, 1986) can be employed to address

the numerical difficulties associated with these edges). In

general, employing non-well-behaved functions is incon-

venient from a numerical implementation perspective.

Failure envelope formulations can be used in two separate

types of practical application: (a) assessment of ultimate

foundation capacity and (b) representing the yield surface

(and the plastic potential for associated flow) in macro-

element models. Failure envelope formulations that are

suitable for both purposes are therefore particularly con-

venient. However, not all of the current failure envelope

formulations are applicable to both applications, either

due to lack of convexity or because they are not well-

behaved. For example, the extensive use of fractional expo-

nents for the HM-based formulations means that the

yield function may not be real-valued in parts of the

domain (e.g. equation (3) is not real-valued for

˜

M,0or

˜

V,0, since raising negative numbers by fractional powers

results in complex values). Furthermore, the composition

approach within this group of formulations results in

singularities in the function and its gradient at

˜

V¼1,

which makes this form of failure envelope inconvenient for

macro-element model implementation. It is noted that the

HM-based formulations were not developed with macro-

element modelling in mind, and thus suitability for this use

should not necessarily be expected. However, failure envel-

opes developed using the formulation framework described

in the current paper are guaranteed to be convex and well-

behaved; they are therefore applicable to a range of practical

applications.

A new framework for convex, well-behaved, failure envelopes

For the purposes of the current paper, a ‘globally convex’

function is defined as a function that is convex throughout

the domain R

n

(real coordinate space of ndimensions, where

nis interpreted in the context of the current work as the

number of individual load and moment components applied

to the foundation). A new framework is proposed for

the formulation of failure envelopes that are guaranteed to

be globally convex and well-behaved for all possible failure

load combinations. Furthermore, the framework is systema-

tic and general. This is significant, as previous researchers

(e.g. Gourvenec, 2007) have suggested that difficulties in

deriving closed-form expressions to approximate failure

envelopes present a practical obstacle to the adoption of

the failure envelope approach in design. The proposed frame-

work provides a systematic process that can be applied

to failure envelope data sets consisting of multiple load

(and moment) components. The framework is also able to

represent various failure envelope shapes that correspond to

different types of shallow foundation. It therefore facilitates

the further development of the failure envelope approach for

practical applications.

In developing this framework, it is recognised that proving

convexity for a general function is typically a difficult task.

For example, a twice differentiable function with a positive

semi-definite Hessian throughout its domain is guaranteed to

be convex. However, a heuristic approach that demonstrates

that a Hessian is positive semi-definite at a finite number of

points is insufficient to prove convexity for the entire domain.

Conversely, demonstrating non-convexity is often relatively

straightforward; a single case where the Hessian is not

positive semi-definite is sufficient.

SURYASENTANA, BURD, BYRNE AND SHONBERG

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The approach employed in the current framework is

therefore to restrict the search for failure envelope formu-

lations to well-behaved polynomial functions that are known

a priori to be globally convex. In developing this procedure,

use is made of a mathematical technique known as ‘sum of

squares’(SOS) programming, which provides a computa-

tionally tractable method for determining globally convex

polynomials.

POLYNOMIAL FAILURE ENVELOPE

FORMULATIONS

General homogeneous polynomials

Ellipsoids are a class of well-behaved functions that are

guaranteed to be convex, as outlined below. A general

ellipsoid can be expressed in the quadratic form

fxðÞ¼ xcðÞ

TAxcðÞ¼1ð4Þ

where x,c[R

n

are vectors and Ais a positive definite square

constant matrix. The Hessian in this case is r2fxðÞ¼2A

and therefore guaranteed to be positive definite. Since posi-

tive definite matrices are also positive semi-definite, it follows

that ellipsoids are convex. As ellipsoids are also well-behaved,

they are candidates for representing failure envelopes, for

which the components of xare the foundation loads (in

an appropriate normalised form) and the components of A

and care selected to provide a best fit with the available

failure load data. However, ellipsoids have the significant

disadvantage in this application that they are not expressive

enough to represent the varied shapes of failure envelopes

that correspond to the range of plausible shallow foundation

configurations.

Ellipsoids are second degree polynomials; higher degree

polynomials provide options for developing failure envelope

functions with increased expressiveness. Since polynomials of

odd degree 3 are guaranteed to be non-convex (Ahmadi

et al., 2013), odd polynomials are excluded as candidates for

the current application. Polynomials of even degree 4 can

either be convex or non-convex, but it is typically computa-

tionally difficult to prove their convexity in the general case

(Ahmadi et al., 2013). Thus, this paper considers a subset of

polynomials of even degree 2, called sum of squares convex

(SOS-convex) polynomials, for which convexity can be

demonstrated in a computationally tractable manner.

Sum of squares polynomials

A polynomial s(x) of degree 2d(where dis a positive

integer) is an SOS polynomial if it can be represented as

sxðÞ¼

X

npoly

j¼1

gjðxÞ2ð5Þ

where g

j

(x) are polynomials of degree 4dand npoly is the

number of individual polynomials, g

j

(x), employed in the

summation. Equivalently, s(x) is SOS if and only if it can be

represented as s(x)¼z

T

Bzwhere Bis a positive semi-definite

matrix and zis a vector of all monomials of degree up

to and including d. This can be shown by noting that

s(x)¼z

T

Bz¼z

T

Q

T

Qz¼(Qz)

T

(Qz)¼g

T

g(where g

T

gis

equation (5) and Q

T

Qis the Cholesky decomposition

of B). It is evident from equation (5) that s(x)0forx[R

n

.

A necessary and sufficient condition for a twice differenti-

able polynomial f(x) to be convex is for its Hessian r2fxðÞto

be positive semi-definite (see Appendix). This requires that

yTr2fðxÞy0 for all x;y[domain of fð6Þ

A subset of polynomials that satisfy equation (6) is defined

by the restricted but more tractable condition

yTr2fðxÞyis SOS for all x;y[domain of fð7Þ

Polynomials f(x) that satisfy equation (7) are convex

(although convex polynomials exist that do not satisfy

equation (7)). The subset of convex polynomials that satisfy

equation (7) is referred to as ‘SOS-convex’polynomials

(Ahmadi & Parrilo, 2012).

A key benefit of employing SOS-convex polynomials

for failure envelope applications is that the requirement

in equation (7) can readily be incorporated within the

search process for failure envelope formulations using

semi-definite programming (Parrilo, 2003). Additionally,

SOS-convex polynomials are well-behaved and support

general n-dimensional loading. Thus, SOS-convex poly-

nomials form the basis of the failure envelope formulations

in the proposed framework.

DETERMINATION OF SOS-CONVEX POLYNOMIAL

FAILURE ENVELOPES

Procedures to determine SOS-convex polynomials to

provide a fit with failure load data sets are outlined below.

Standardise the data

Failure envelope data typically consist of sets of discrete

failure load combinations. These failure load combinations

are standardised by

ˉ

xi¼xixi;c

xi;ref

ð8Þ

where x

i

is a load component, x

i,ref

is a reference load and x

i,c

is a shift parameter. The purpose of equation (8) is to shift

and scale the data such that the values of ˉ

xifor uniaxial

loading in the positive and negative directions (according to

the selected coordinate system) are 1 and 1, respectively.

Geometrically, this ensures that the failure envelope inter-

sects each ˉ

xiaxis at ± 1. Importantly, if fˉ

xðÞis convex, then

convexity in f(x) is preserved as the mapping in equation (8)

is affine (Boyd & Vandenberghe, 2004).

Define the failure envelope functional form

SOS-convex polynomial functions of the standardised

loading variables ˉ

x, of even degree 2dwhere dis a positive

integer, are now sought to represent the failure envelope fˉ

xðÞ.

Although SOS-convex polynomials are in general non-

homogeneous, homogeneous forms are adopted in the

current framework as they have the important advantage

that the coefficients for uniaxial loading can be identified

straightforwardly, as indicated in step 3 below.

As an example, a homogeneous polynomial fˉ

xðÞ with

degree 2d= 4 and number of loading dimensions n¼2, with

coefficients a

i

(that are subsequently determined to ensure

that the polynomial is SOS-convex) is

fðˉ

xÞ¼fˉ

x1;ˉ

x2

ðÞ

¼ˉ

x4

1a1þˉ

x3

1x2a2þˉ

x2

1

ˉ

x2

2a3þˉ

x1

ˉ

x3

2a4þˉ

x4

2a5ð9Þ

The failure envelope is defined by fˉ

xðÞ1¼0.

Apply the uniaxial conditions

The standardisation process in step 1 requires that the

coefficients of all monomials comprising a single loading

A SYSTEMATIC FRAMEWORK FOR FORMULATING CONVEX FAILURE ENVELOPES 3

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variable are 1. (This is to represent failure correctly

for uniaxial loading conditions.) For the example in

equation (9), this condition gives a

1

¼1 and a

5

¼1.

Apply symmetry principles

Other coefficients can be determined by exploiting

any available symmetry conditions. For the example in

equation (9), if the problem is symmetric such that

fˉ

x1;ˉ

x2

ðÞ¼fˉ

x1;ˉ

x2

ðÞ, then the coefficients of all monomials

that are odd in ˉ

x1must be zero; in the current example this

implies a

2

¼0 and a

4

¼0.

Set up a convex optimisation problem

To identify the remaining unknown coefficients (there is

only one unknown a

3

in the current example but a greater

number of coefficients will typically need to be determined at

this stage), a convex optimisation problem is set up. This is

based on the conditions: (a)fˉ

xðÞis SOS-convex and (b)fˉ

xðÞ

provides a best fit with the failure envelope data by

minimising the objective function C

C¼X

ndata

i¼1

fˉ

xdata

i

1

2ð10Þ

where ˉ

xdata

iis a set of failure load combinations and ndata is

the size of the data set. The optimisation problem is therefore

minimise C

subject to ˉ

yTr2fˉ

xðÞ

ˉ

yis SOS for all ˉ

x;ˉ

y[domain of fðˉ

xÞ

ð11Þ

To solve the SOS program defined in equation (11), it must

first be converted into an equivalent semi-definite program.

In the current work, the Matlab toolbox ‘YALMIP’

(Löfberg, 2004) was used to convert SOS problems such as

equation (11) automatically to their equivalent semi-definite

programs (Löfberg, 2009). Thereafter, the unknown coeffi-

cients in fðˉ

xÞcan be readily determined using this toolbox.

EXAMPLE APPLICATIONS

Three examples are provided below to demonstrate the

application of the proposed formulation framework. The first

example concerns the failure envelope for a surface circular

footing on homogeneous clay; the second example is the

failure envelope for a suction caisson foundation embedded

in homogeneous clay; the last example is a new failure

envelope formulation to represent Model B, which demon-

strates that an existing failure envelope formulation can be

approximated within the proposed framework to achieve the

advantages of global convexity and well-behaved function

attributes.

In all the examples, YALMIP was used, in conjunction

with the SeDuMi semi-definite solver (Sturm, 1999), to solve

the SOS programs. All the failure envelopes for the examples

are plotted in the normalised load space (e.g.

˜

H

˜

M). This

enables convenient comparison with figures in previous

publications (for the case of Model B).

Surface and caisson foundations

The first two examples use the VHM failure load

combinations for surface and caisson foundations in homo-

geneous clay, determined as described in Suryasentana et al.

(2019) using the ‘sequential swipe test’in three-dimensional

(3D) finite-element analysis. The soil in these cases was

modelled as a linear elastic–perfectly plastic von Mises

material. Fig. 1 shows the foundation configurations being

considered, including the assumed location of the loading

reference point (LRP) (which is needed to provide a datum

for the definition of the applied moment).

Based on the numerical failure load data, standardised

forms of the failure load data ½

ˉ

H;

ˉ

M;

ˉ

Vare obtained, from

equation (8), with the computed uniaxial capacities [H

0

,M

0

,

V

0

] adopted as the reference values and [H

c

,M

c

,V

c

]¼[0,

0, 0] as the shift parameters. The foundations analysed

in Suryasentana et al. (2019) assumed that the soil and

foundation were fully bonded (i.e. no contact breaking) and

thus, the foundation has the same tension and compression

vertical uniaxial capacities, resulting in V

c

¼0 in the current

example.

Two separate SOS-convex polynomial failure envelopes

have been determined: f

4

(with degree 2d¼4) and f

6

(with

degree 2d¼6). The general forms of these polynomials are

f4

ˉ

H;

ˉ

M;

ˉ

V

¼a1

ˉ

H4þa2

ˉ

H3ˉ

Mþa3

ˉ

H2ˉ

M2þa4

ˉ

H

ˉ

M3

þa5

ˉ

M4þa6

ˉ

H3ˉ

Vþa7

ˉ

H2ˉ

M

ˉ

V

þa8

ˉ

H

ˉ

M2ˉ

Vþa9

ˉ

M3ˉ

Vþa10

ˉ

H2ˉ

V2

þa11

ˉ

H

ˉ

M

ˉ

V2þa12

ˉ

M2ˉ

V2

þa13

ˉ

H

ˉ

V3þa14

ˉ

M

ˉ

V3þa15

ˉ

V4

ð12Þ

f6

ˉ

H;

ˉ

M;

ˉ

V

¼a1

ˉ

H6þa2

ˉ

H5ˉ

Mþa3

ˉ

H4ˉ

M2

þa4

ˉ

H3ˉ

M3þa5

ˉ

H2ˉ

M4þa6

ˉ

H

ˉ

M5

þa7

ˉ

M6þa8

ˉ

H5ˉ

Vþa9

ˉ

H4ˉ

M

ˉ

V

þa10

ˉ

H3ˉ

M2ˉ

Vþa11

ˉ

H2ˉ

M3ˉ

V

þa12

ˉ

H

ˉ

M4ˉ

Vþa13

ˉ

M5ˉ

Vþa14

ˉ

H4ˉ

V2

þa15

ˉ

H3ˉ

M

ˉ

V2þa16

ˉ

H2ˉ

M2ˉ

V2

þa17

ˉ

H

ˉ

M3ˉ

V2þa18

ˉ

M4ˉ

V2þa19

ˉ

H3ˉ

V3

þa20

ˉ

H2ˉ

M

ˉ

V3þa21

ˉ

H

ˉ

M2ˉ

V3þa22

ˉ

M3ˉ

V3

þa23

ˉ

H2ˉ

V4þa24

ˉ

H

ˉ

M

ˉ

V4þa25

ˉ

M2ˉ

V4

þa26

ˉ

H

ˉ

V5þa27

ˉ

M

ˉ

V5þa28

ˉ

V6

ð13Þ

Application of the uniaxial condition gives a

1

,a

5

,a

15

¼1

in f

4

and a

1

,a

7

,a

28

¼1inf

6

. Additionally, symmetry in

ˉ

H

and

ˉ

Mrequires that a

6

,a

7

,a

8

,a

9

,a

13

,a

14

¼0inf

4

and a

8

,a

9

,

a

10

,a

11

,a

12

,a

13

,a

19

,a

20

,a

21

,a

22

,a

26

,a

27

¼0inf

6

. The

remaining coefficients are determined from the optimisation

in equation (11).

MSurface

foundation

LRP H

V

Caisson skirt

(when present)

Fig. 1. VHM loading configuration for surface foundation and caisson

SURYASENTANA, BURD, BYRNE AND SHONBERG4

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Results. The coefficients for f

4

and f

6

determined from this

process are listed in Tables 1 and 2, respectively. Figs 2 and 3

compare the normalised VH,VM and HM failure envelopes

determined by f

4

1¼0 and f

6

1¼0 for the surface and

caisson foundation, respectively; also shown are the corre-

sponding 3D finite-element results from Suryasentana et al.

(2019). Differences between f

4

and f

6

can be seen more clearly

in Fig. 4 (which omits the finite-element results).

Both f

4

and f

6

are seen to provide reasonably good

approximations of the computed data, although there is a

small under-prediction of the VM failure envelope for the

suction caisson (Fig. 3(b)) and f

4

is unconservative compared

to f

6

for the surface foundation at high vertical loads

(Fig. 2(c)). It is seen that f

6

provides a better fit with the

numerical data than f

4

for the surface foundation (Fig. 2(c)),

but f

4

is marginally better for the caisson (Fig. 3(c)). This is

confirmed by the data in Table 3, which shows values of the

objective function (and their root-mean) at the end of the

optimisation process (the smaller the better).

Model B

Model A, introduced in the earlier section ‘Existing failure

envelope formulations’, is representative of a class of func-

tions used by numerous authors to represent yield envelopes.

A modified form of this model termed ‘Model B’, found to

provide an improved fit to experimental data of spudcan

performance on clay, is described in Martin (1994) and

Martin & Houlsby (2000). The Model B formulation is

f

˜

H;

˜

M;

˜

V

¼

˜

H2þ

˜

M22e

˜

H

˜

Mβ2˜

V2β1ð1

˜

VÞ2β2¼0

ð14Þ

Table 1. Best-fit coefficients in f

4

for the failure envelope data from

Suryasentana et al. (2019)

Foundation a

2

a

3

a

4

a

10

a

11

a

12

Surface 0·36 0·9 1·43 0·4 0·84 1·64

Caisson 3·55 5·12 3·51 1·72 3·59 2·07

Table 2. Best-fit coefficients in f

6

for the failure envelope data from Suryasentana et al. (2019)

Foundation a

2

a

3

a

4

a

5

a

6

a

14

a

15

a

16

a

17

a

18

a

23

a

24

a

25

Surface 0·33 1·22 2·17 2·34 1·86 0·03 0·34 1·1 0·29 0·84 1·97 1·52 4·72

Caisson 5·18 11·9 15·45 11·96 5·2 2·72 10·86 17·27 13·23 4·21 2·13 2·93 1·04

1·0

0·5

–0·5

–1·0

–1·0 –0·5 0

(a) (b)

(c)

0·5 1·0

0

H

˜

V

˜

–1·0 –0·5 0 0·5 1·0

V

˜

1·0

0·5

–0·5

–1·0

0

M

˜

–1·0–1·5 –0·5 0 0·5 1·0 1·5

H

˜

1·0

0·5

–0·5

–1·0

0

M

˜

3DFE

f4

f6

3DFE

f4

f6

3DFE

f4

f6

Increasing V

˜

Fig. 2. Comparison of the normalised VH,VM and HM failure envelopes predicted by f

4

and f

6

with 3D finite-element (denoted 3DFE in figure)

results (Suryasentana et al., 2019) for a surface foundation. The HM envelopes correspond to normalised vertical loads:

˜

V= 0, 0·25, 0·5, 0·625,

0·75, 0·875

A SYSTEMATIC FRAMEWORK FOR FORMULATING CONVEX FAILURE ENVELOPES 5

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or, alternatively

fð

˜

H;

˜

M;

˜

VÞ¼ð

˜

H2þ

˜

M22e

˜

H

˜

MÞ1=2β2

ˉ

β1=β2˜

Vβ1=β2ð1

˜

VÞ

¼0

ð15Þ

where e¼0518 þ118

˜

Vð

˜

V1Þ,β¼β1þβ2

ðÞ

β1þβ2

ðÞ

=ββ1

1ββ2

2,

and β

1

¼0·764 and β

2

¼0·882 are parameters, selected to

fit the model to the data. It is instructive to use the new

failure envelope framework to develop an alternative version

1·0

0·5

–0·5

–1·0

0

H

˜

1·0

0·5

–0·5

–1·0

0

M

˜

M

˜

H

˜

–1·0 –0·5 0

(a)

0·5 1·0

V

˜

–1·0 –0·5 0

(b)

(c)

0·5 1·0

V

˜

2

1

0

–1

–2

–2 –1 0 1 2

3DFE

f4

f6

3DFE

f4

f6

3DFE

f4

f6

Increasing V

˜

Fig. 3. Comparison of the normalised VH,VM and HM failure envelopes predicted by f

4

and f

6

with the 3D finite-element results (Suryasentana

et al., 2019) for a suction caisson foundation. The HM envelopes correspond to normalised vertical loads:

˜

V= 0, 0·25, 0·5, 0·625, 0·75, 0·875

1·0

0·5

–0·5

–1·0

0

M

˜

–1·0–1·5 –0·5 0

(a)

0·5 1·0 1·5

M

˜

H

˜

H

˜

(b)

2

1

0

–1

–2

–2 –1 0 1 2

f4

f6

f4

f6

Increasing V

˜

Increasing V

˜

Fig. 4. Comparison of the normalised HM failure envelopes predicted by f

4

and f

6

under increasing vertical loading (i.e.

˜

V= 0, 0·25, 0·5, 0·625,

0·75, 0·875): (a) surface foundation; (b) suction caisson foundation

Table 3. Minimised objective values (and corresponding root-mean

values) for f

4

and f

6

at the end of the optimisation process for the

surface and caisson foundation results from Suryasentana et al. (2019)

Foundation Cfor f

4

Cfor f

6

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

C=ndata

pfor f

4

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

C=ndata

pfor f

6

Surface 36·96 31·05 0·270 0·248

Caisson 20·3 20·35 0·246 0·247

Note: ndata is the number of sets of failure load combinations

(ndata = 506 and ndata = 335 for the surface and caisson foundation,

respectively).

SURYASENTANA, BURD, BYRNE AND SHONBERG6

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of Model B, to demonstrate how the new framework can be

applied to existing formulations.

Failure envelope data were generated using equation (15)

in the form of the

˜

V

˜

Hfailure envelope,

˜

V

˜

Mfailure envelope

and the

˜

H

˜

Mfailure envelopes for

˜

V= 0·125, 0·25, 0·375, 0·5,

0·625, 0·75, 0·875. The data were standardised according

to equation (8), where the adopted shift parameters

were ½

˜

Hc;

˜

Mc;

˜

Vc¼ ½0;0;05. Thus, ½

˜

Href ;

˜

Mref ;

˜

Vref ¼

½0995;0995;05(the failure envelope intersects the

˜

Hand

˜

Maxes at

˜

H= ±0·995 and

˜

M¼+0995 when

˜

V¼

˜

Vc¼

05, from equation (15)).

Failure envelopes have been determined on the basis of f

4

and f

6

(i.e. equations (12) and (13)) incorporating the same

uniaxial and symmetry conditions as adopted in the earlier

section ‘Surface and caisson foundations’examples. The

remaining coefficients were determined using the optimis-

ation in equation (11).

The computed coefficients are listed in Tables 4 and 5. Fig. 5

shows the normalised VH,VM and HM failure envelopes for

f

4

1¼0 and f

6

1¼0, together with comparisons with the

original Model B equation. Evidently, both f

4

and f

6

provide

reasonably good approximations of the Model B equation

(although neither of them predicts the correct value of

˜

Vat

which the peaks in the

˜

V

˜

Hand

˜

V

˜

Menvelopes occur). Table 6

shows the minimised objective function values (and their

root-mean) for f

4

and f

6

at the end of the optimisation; these

data suggest that f

4

provides a better fit than f

6

. Figs 5(a)

and 5(b) show that, at

˜

V¼0and

˜

V¼1, the failure envelopes

implied by f

4

and f

6

are more ‘rounded’than Model B, and the

surface normal is directed along the

ˉ

Vaxis. This means that,

when employed as a plastic potential, the model correctly

predicts vertical plastic displacement for pure vertical loading;

this is significant, as special numerical procedures are required

to achieve such behaviour with Model B (Martin, 1994). In

general, the results show that the SOS-convex Model B

formulation provides a failure envelope that is reasonably

close to the formulation in equation (15), although it is evident

from Fig. 5(c) that the SOS-convex formulation is unconser-

vative for relatively high values of normalised vertical load; the

SOS-convex formulation, however, is guaranteed to be

globally convex and well-behaved.

Table 4. Best-fit coefficients in f

4

for the Model B failure envelope

data

a

2

a

3

a

4

a

10

a

11

a

12

0·86 2 0·86 3·48 2·93 3·48

Table 5. Best-fit coefficients in f

6

for the Model B failure envelope data

a

2

a

3

a

4

a

5

a

6

a

14

a

15

a

16

a

17

a

18

a

23

a

24

a

25

1·35 3·59 2·74 3·59 1·35 3·56 4·94 8·94 4·94 3·56 7·37 6·59 7·37

1·0

0·5

0

–0·5

–1·0

H

˜

1·0

0·5

0

–0·5

–1·0

M

˜

00·2

(a)

0·60·4 0·8 1·0

V

˜

00·2

(b)

0·60·4 0·8 1·0

V

˜

1·0

0

0·5

–0·5

–1·0

M

˜

–0·5–1·0 0

(c)

0·5 1·0

H

˜

Model B

f4

f6

Model B

f4

f6

Model B

f4

f6

Increasing V

˜

Fig. 5. Comparison of the normalised VH,VM and HM failure envelopes predicted by f

4

and f

6

with the original Model B equation. The HM

envelopes correspond to normalised vertical loads:

˜

V= 0·5, 0·75, 0·875

A SYSTEMATIC FRAMEWORK FOR FORMULATING CONVEX FAILURE ENVELOPES 7

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Macro-element model for a surface foundation

To demonstrate the application of the proposed SOS-

convex failure envelope formulations in a macro-element

model, the 4th degree version of the failure envelope

(equation (12)) was employed, using the coefficients for a

surface foundation in Table 1. Failure envelopes of the form

f

ˉ

H;

ˉ

M;

ˉ

V

such as equation (12) are implicit functions of

the loading variables H,M,Vby way of the affine trans-

formation in equation (8) (in this case,

ˉ

H¼H=H0,

ˉ

M¼

M=M0and

ˉ

V¼V=V0). The failure envelope f

4

(H,M,

V)1¼0 was implemented as the yield surface and plastic

potential of a macro-element model, with the elasto-plastic

integration performed using the implicit closest point

projection method (Simo & Hughes, 2006). The uniaxial

capacities (V

0

,H

0

,M

0

) were determined from the finite-

element data in Suryasentana et al. (2019), while the elastic

stiffness matrix for the macro element was calibrated using

pre-failure data obtained from the finite-element results.

The macro-element model is used, as an example, to

simulate a sequential swipe test in the HM load space for

planar HM loading with

˜

V= 0·25, to replicate the corres-

ponding finite-element simulations in Suryasentana et al.

(2019). Fig. 6 provides a comparison of the load–

displacement behaviour and the normalised HM failure

envelope, computed using the macro-element model and the

3D finite-element model (Suryasentana et al., 2019). It is

evident that the macro-element predictions agree well with

the finite-element results.

DISCUSSION

The procedures presented in the paper provide a con-

venient means of generating failure envelopes that are convex

and well-behaved. This is achieved by selecting functions

from a restricted set of SOS-convex polynomials. The process

is demonstrated by applying it to previously published

numerical data on failure load combinations for a surface

footing and a caisson on soil that is modelled as an elastic–

perfectly plastic von Mises material. Additionally, it is used

to reformulate a previously determined failure envelope,

Model B.

The results indicate that an SOS-convex polynomial of

degree 6, f

6

, does not necessarily provide a better fit to a

prescribed data set than an SOS-convex polynomial of

degree 4, f

4

. The results indicate that f

6

is a better fit for the

surface foundation failure envelope than f

4

, but the opposite is

true for the caisson and Model B failure envelopes. This may

seem counterintuitive, but it is a consequence of the fact that

Table 6. Minimised objective values (and corresponding root-mean

values) for f

4

and f

6

at the end of the optimisation process for the

Model B example

Foundation Cfor f

4

Cfor f

6

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

C=ndata

p

for f

4

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

C=ndata

p

for f

6

Model B 8·23 11·22 0·0476 0·0556

Note: ndata = 3634 is the number of sets of failure load combinations.

1·0

0·5

0

–0·5

–1·0

H

˜

V

˜ = 0·25

1·0

0·5

0

–0·5

–1·0

M

˜

1·0

0·5

0

–0·5

–1·0

M

˜

00·025 0 0·02 0·04 0·06 0·08 0·10 0·12

(a)

0·0750·050 0·100 0·125 0·150

Sh/D θm

Macro-element

3DFE

Macro-element

3DFE

Macro-element

3DFE

–0·5–1·0 0

(c)

(b)

0·5 1·0

H

˜

Fig. 6. Comparison of load–displacement and the normalised HM failure envelopes predicted by the macro-element and 3D finite-element results

(Suryasentana et al., 2019), for a sequential swipe test in the HM load space when

˜

V= 0·25. S

h

and θ

m

refer to the horizontal and rotational

displacements of the surface foundation, respectively. Dis the diameter of the surface foundation

SURYASENTANA, BURD, BYRNE AND SHONBERG8

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the current approach is restricted to the use of homogeneous

polynomials. Although a general polynomial can be used to

represent any polynomial of lower degree, this is not the case

for the homogeneous polynomials employed here. For

practical applications of this method, it is recommended that

a relatively low degree homogeneous polynomial (e.g. degree=

4) is initially adopted. Higher degree polynomials would then

be considered if the low degree polynomial is found to provide

an unsatisfactory fit to the data.

The YALMIP toolbox provides a useful SOS decompo-

sition feature to double check that the failure envelopes are,

indeed, convex throughout the entire domain. For example,

for f

4

employed for Model B, the Hessian can be decomposed

into SOS form using YALMIP as

where ˉ

x¼½

ˉ

V;

ˉ

H;

ˉ

MT,ˉ

y¼½

ˉ

V1;

ˉ

H1;

ˉ

M1T[domain of f

4

.

This equation demonstrates that ˉ

yTr2f4

ˉ

x

ðÞ

ˉ

yis SOS, hence

ˉ

yTr2f4

ˉ

xðÞ

ˉ

y0 for its entire domain; f

4

is therefore shown to

be SOS-convex.

The values of the coefficients in the failure envelope

polynomials determined using the proposed framework do

not necessarily have a physical interpretation. This disadvan-

tage is outweighed by the new approach being able to represent

a wide range of failure envelope shapes, while maintaining

desirable theoretical attributes. A simple way to interpret the

polynomial coefficients would be to view their (absolute)

magnitudes as indicative of the strength of coupling between

the respective loading components at failure.

The framework presented in the paper has certain limit-

ations, as follows. The application of the SOS-convex failure

envelope formulation to macro-element modelling has only

been demonstrated for perfect plasticity with associated flow.

Further work is needed to develop the modelling approach for

the case of non-associated flow and hardening plasticity.

Furthermore, the current study only explores the application

of homogeneous polynomials. Non-homogeneous SOS-

convex polynomials are not explored, but they potentially

provide greater expressiveness than homogeneous SOS-convex

polynomials (since homogeneous polynomials are subsets of

non-homogeneous polynomials). However, based on initial

analyses using non-homogeneous polynomials, the SeDuMi

solver employed in the current work encountered convergence

problems for some of the above examples; this contrasts with

the homogeneous polynomials where convergence issues did

not occur. Further work is needed to explore the implemen-

tation of non-homogeneous SOS-convex polynomials and

their potential benefits.

CONCLUSIONS

This paper introduces a novel SOS-convex polynomial-

based framework to systematically formulate failure envel-

opes that are suitable for use in both ultimate capacity

calculations and macro-element modelling. The approach is

applicable to failure envelope data sets of multiple loading

dimensions, and it is demonstrated in the current paper for

VHM loading. The principal advantage of the proposed

framework is that it leads to formulations for failure envelope

functions (and yield functions) that are guaranteed to be

globally convex and well-behaved. As discussed in this paper,

these attributes lead to significant advantages in terms of

theoretical considerations as well as the numerical stability of

macro-element models.

Examples are presented to demonstrate the application of

the framework to three different shallow foundation cases,

with significantly different failure envelope shapes. These

examples demonstrate the expressiveness and general capa-

bilities of the proposed framework. The framework ration-

alises and expedites the formulation of failure envelopes and

therefore supports the development of the failure envelope

approach for design applications.

Although not shown here, this framework may have

potential uses in other fields of solid mechanics, such as

determining convex yield functions from experimentally

determined yield data for different materials.

ACKNOWLEDGEMENT

The first author would like to acknowledge the generous

support of Ørsted Wind Power for funding his DPhil

studentship at the University of Oxford.

APPENDIX

Convex functions

A function fis defined to be convex if its domain is a convex set

and

fθxþ1θðÞy½θfðxÞþ 1θðÞfðyÞð17Þ

for all x,y[domain of fand for 0 θ1 (Boyd & Vandenberghe,

2004). Depending on the differentiability of f, there are two

equivalent conditions to equation (17) that characterise the

convexity of f.

If fis differentiable (i.e. its gradient rfðxÞexists everywhere in its

domain), fis convex if its domain is a convex set and

fðyÞfðxÞþrfðxÞTyxðÞ ð18Þ

for all x,y[domain of f. Equation (18) is known as the first-order

condition (Boyd & Vandenberghe, 2004).

If fis twice differentiable (i.e. its Hessian r2fðxÞexists everywhere

in its domain), fis convex if its domain is a convex set and r2fðxÞis

positive semi-definite everywhere in its domain –that is

yTr2fðxÞy0ð19Þ

ˉ

yTr2f4

ˉ

xðÞ

ˉ

y¼32835

ˉ

V

ˉ

V130493

ˉ

M

ˉ

M1þ09486

ˉ

M

ˉ

H1þ09486

ˉ

H

ˉ

M130493

ˉ

H

ˉ

H1

2

þ20827

ˉ

V

ˉ

M1þ20827

ˉ

V

ˉ

H120827

ˉ

M

ˉ

V1þ20827

ˉ

H

ˉ

V1

2

þ11788

ˉ

V

ˉ

M111788

ˉ

V

ˉ

H111788

ˉ

M

ˉ

V111788

ˉ

H

ˉ

V1

2

þ15127

ˉ

M

ˉ

M1þ15127

ˉ

H

ˉ

H1

2þ13083

ˉ

M

ˉ

H1þ13083

ˉ

H

ˉ

M1

2

þ00816

ˉ

V

ˉ

V1þ03209

ˉ

M

ˉ

M1þ11727

ˉ

M

ˉ

H1þ11727

ˉ

H

ˉ

M1þ03209

ˉ

H

ˉ

H1

2

þ07908

ˉ

V

ˉ

M1þ07908

ˉ

V

ˉ

H107908

ˉ

M

ˉ

V107908

ˉ

H

ˉ

V1

2

þ07792

ˉ

V

ˉ

M107792

ˉ

V

ˉ

H107792

ˉ

M

ˉ

V1þ07792

ˉ

H

ˉ

V1

2

þ11009

ˉ

V

ˉ

V1þ05572

ˉ

M

ˉ

M101142

ˉ

M

ˉ

H101142

ˉ

H

ˉ

M1þ05572

ˉ

H

ˉ

H1

2

ð16Þ

A SYSTEMATIC FRAMEWORK FOR FORMULATING CONVEX FAILURE ENVELOPES 9

Downloaded by [] on [30/04/19]. Published with permission by the ICE under the CC-BY license

for all x,y[domain of f. This is known as the second-order

condition (Boyd & Vandenberghe, 2004) and is the principal

condition for assessing the convexity of functions in the current

paper. One way to determine whether a matrix is positive

semi-definite is to check whether all of its eigenvalues are

non-negative everywhere in its domain.

An example of a twice differentiable, convex function is the von

Mises yield function, where it can be shown that the eigenvalues of

the Hessian are always non-negative. The Tresca yield function is

convex according to the general condition in equation (17), but since

it is not differentiable over its entire domain (e.g. at the ‘corners’)

then the more restrictive first- and second-order conditions do not

apply.

A yield surface should not be confused with a yield function. A

yield surface is usually defined as the zero level set of ayield function

f(x) (i.e. {x|f(x)¼0}). A convex yield surface does not necessarily

imply convexity of f(x). Conversely, if f(x) is convex, then the

yield surface is necessarily convex. These conditions arise from

convex analysis relating convex functions and their sublevel sets

(Boyd & Vandenberghe, 2004), where a k-sublevel set is defined

as {x|f(x)k}. In continuum plasticity, the 0-sublevel set of a yield

function typically represents the set of admissible stress states.

NOTATION

Apositive definite matrix representing an ellipsoid in quadratic

form

Bpositive semi-definite matrix representing coefficients of an

SOS polynomial

ccentre coordinates of an ellipsoid

Ddiameter of surface foundation

dpositive integer representing half the degree of an SOS

polynomial

g

j

polynomial components of an SOS polynomial

Hhorizontal load

H

0

horizontal uniaxial capacity

˜

Hnormalised horizontal load

ˉ

Hstandardised horizontal load

Mmoment

M

0

moment uniaxial capacity

ˉ

Mstandardised moment load

˜

Mnormalised moment load

nnumber of loading dimensions

Qmatrix from Cholesky decomposition of B

R

n

real coordinate space of ndimensions

S

h

horizontal displacement of surface foundation

sSOS polynomial

Vvertical load

V

0

vertical uniaxial capacity

ˉ

Vstandardised vertical load

˜

Vnormalised vertical load

x

i

load component

x

i,c

shift parameter

x

i,ref

reference load

zvector of monomials up to and including degree d

θ

m

rotational displacement of surface foundation

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